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Note that \(G=IX\), \(G'=\overline{[G,G]}\) and \([G,G]\) is generated by elements of form \((\alpha x)(\beta y)(\alpha x)^{-1}(\beta y)^{-1} =(\alpha \bullet x)(\alpha \beta \bullet x)^{-1}(\alpha \beta \bullet y)(\beta \bullet y)^{-1}\), where \(\alpha ,\beta \in I\) and \(x,y\in X\). By taking \(\alpha \in I\) to be the preimage of \(\gamma \in \Gamma \), and \(\beta =1\), we know that “\(\supset \)” holds.

Note that RHS is a compact subgroup (it is the image of a compact set, \((\gamma \bullet x)x^{-1}(\gamma \bullet y)y^{-1} =(\gamma \bullet xy)(xy)^{-1}\), and \(((\gamma \bullet x)x^{-1})^{-1} =(\gamma \bullet x^{-1})x\)). Therefore, to prove “\(\subset \)”, we only need to show that \((\alpha \bullet x)x^{-1}\) is in RHS for all \(\alpha \in I\) and \(x\in X\).

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Class field theory.

Since \(L_n\) is maximal.

Since \(K_{n+1}\) and \(L_n\) are Galois over \(K_n\), the \(K_{n+1}L_n\) is also Galois over \(K_n\), and the natural map \({\mathrm{Gal}}(K_{n+1}L_n/K_{n+1})\to {\mathrm{Gal}}(L_n/K_n)\) is injective. ???

Class field theory.

The inverse map maps \((\sigma _n)_{n\geq 0}\) to \(x\mapsto \sigma _n(x)\) if \(x\in L_n\).

The \(\Gamma _n\)-action on \(X_n\) is given by Definition 5.1. The \(\Gamma \)-action on \(X_\infty \) is similar.

Class field theory.

Let \(D_{\mathfrak {l}}\) be the decomposition subgroup of \({\mathfrak {l}}\) in \(\Gamma :={\mathrm{Gal}}(K_\infty /K)\cong {\mathbb {Z}}_p\), which is a closed subgroup. If \({\mathfrak {l}}\) is an infinite place then \(D_{\mathfrak {l}}\) is finite, hence must be trivial. In the following assume \({\mathfrak {l}}\) is a finite place. Let \(l=\operatorname{char}({\mathcal{O}}_K/{\mathfrak {l}})\neq p\), then \(K_{\mathfrak {l}}/{\mathbb {Q}}_l\) is a finite extension, and let \((K_\infty )_{\mathfrak {l}}:=\varinjlim (K_n)_{\mathfrak {l}}\), then \(D_{\mathfrak {l}}={\mathrm{Gal}}((K_\infty )_{\mathfrak {l}}/K_{\mathfrak {l}})\) is a subgroup of \({\mathbb {Z}}_p\).

We have \(K_{\mathfrak {l}}\subset K_{\mathfrak {l}}^{\mathrm{unr}}\subset K_{\mathfrak {l}}^{\mathrm{ab}}\), and

So \(K_{\mathfrak {l}}\) has at least one \({\mathbb {Z}}_p\)-extension, i.e. the unique unramified \({\mathbb {Z}}_p\)-extension. If there are other \({\mathbb {Z}}_p\)-extensions of \(K_{\mathfrak {l}}\), then there exists a Galois extension of \(K_{\mathfrak {l}}\) with Galois group isomorphic to \({\mathbb {Z}}_p^2\).

However, \({\mathrm{Gal}}(K_{\mathfrak {l}}^{\mathrm{ab}}/K_{\mathfrak {l}}^{\mathrm{unr}})\) doesn’t have a quotient isomorphic to \({\mathbb {Z}}_p\), because by local class field theory, \({\mathrm{Gal}}(K_{\mathfrak {l}}^{\mathrm{ab}}/K_{\mathfrak {l}}^{\mathrm{unr}})\cong {\mathcal{O}}_{K_{\mathfrak {l}}}^\times \cong (\text{a finite group})\times {\mathbb {Z}}_l^{[K_{\mathfrak {l}}:{\mathbb {Q}}_l]}\) which is a finite group times a pro-\(l\) group, obviously it doesn’t have a quotient isomorphic to \({\mathbb {Z}}_p\). So there is only one \({\mathbb {Z}}_p\)-extension of \(K_{\mathfrak {l}}\), note that \((K_\infty )_{\mathfrak {l}}/K_{\mathfrak {l}}\) is either trivial or a \({\mathbb {Z}}_p\)-extension, in both cases it must be contained in \(K_{\mathfrak {l}}^{\mathrm{unr}}\).

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The \(Y\subset X_\infty \) is constructed as follows. Let \({\mathfrak {p}}_1,\cdots ,{\mathfrak {p}}_s\) be primes of \({\mathcal{O}}_K\) which are (totally) ramified in \(K_\infty \) (they are all above \({\mathfrak {p}}\) by Lemma 5.15). For each \({\mathfrak {p}}_i\) fix a place of \(L_\infty \) above it and let \(I_i\subset G\) be its inertia subgroup. Since \(L_\infty /K_\infty \) is unramified and \(K_\infty /K\) is totally ramified at \({\mathfrak {p}}_i\), we have \(I_i\cap X_\infty =1\), and \(I_i\hookrightarrow G/X_\infty =\Gamma \) is surjective, hence bijective. Therefore \(G=I_iX_\infty =X_\infty I_i\). For each \(i\) let \(\sigma _i\in I_i\) be the preimage of \(\gamma \in \Gamma \), then \(\sigma _i\) is a topological generator of \(I_i\). For \(i=2,\cdots ,s\), let \(a_i\in X_\infty \) be some element such that \(\sigma _i=a_i\sigma _1\in X_\infty I_1=G\). Take \(Y\) to be the closure of the \(\Lambda \)-submodule generated by \(a_2,\cdots ,a_s\) and \(\{ (\gamma \bullet x)x^{-1}\mid x\in X_\infty \} \).

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